A category is called leftif it has a strict initial object. Consider a left category C with a strict initial 0. A map is called non-initialif its domain is not initial. A class of objects (or a full subcategory) B of C is called unidense if for any non-initial object X in C there is a map from a non-initial object in B to X. By a full left dense subcategoryof C we mean a full dense subcategory of C containing 0. Note that any left dense full subcategory of C is unidense.

If D is another category with a strict initial object, a functor F: D --> C is called nondegenerateif for any object X in D, F(X) is initial iff X is initial.

Definition 1. (a) A non-initial object T is called unisimpleif for any two non-initial maps f: X --> T and g: Y --> T there are two non-initial maps r: R --> X and s: R --> Y such that fr = gs (cf. [Luo 1998, (3.3.5)]). (b) C is called atomic if the class of unisimple objects is unidense. (c) C is called unisimpleif any non-initial object is unisimple.

Denote by S(C) the full subcategory of unisimple objects of C. Adding the initial object 0 to S(C) we obtain a category S*(C) with 0 as a strict initial object. Note that in general S*(C) is not unisimple.

Proposition 2. (a) Any unisimple category is atomic. (b)If C is atomic then S*(C) is unisimple. (c) C is atomic if it has a full unidense atomic subcategory. (d) C is atomic if it has a full left dense atomic subcategory.

Proof. (a) and (d) are obvious; (b) can be verified directly. (c) If B is a full unidense atomic subcategory of C then S(B) ÍS(C). Since B is atomic, S(B) is unidense in B.By assumption B is unidense in C, thus S(B) is unidense in C. It follows that S(C) is also unidense in C. (e) follows from (c) and (d).

Denote by SET the metacategory of sets. Let S*: S*(C) --> SET be the functor sending 0 to the empty set and each non-initial object in S*(C) to a one point set. Let kC: C --> SET be the Kan extension of S* to C. The functor kC is uniquely determined by C up to equivalence. If kC(X) is small for each object X in C then kC is regarded as a functor from C to the category Set of small sets.

Example 2.1. For any object X one can define kC(X) directly: an element of kC(X) is represented by a map p: P -->X from a unisimple object P to X. If q: Q -->X is another such map then p and q represent the same element of kC(X) iff there are two maps r: R --> P and s: R --> Q such that pr = qs.

Proposition 3.C is atomic iff kC is nondegenerate.

Proof. By (2.1) kC(X) is non-empty iff there is a map from a unisimple object to X. This implies that S(C) is unidense iff kC is nondegenerate.

Theorem 4. If B is a full unidense atomic subcategory of C then C is atomic and kC is the Kan extension of kD.

Proof. The first assertion has been noticed in (2.c). For the second assertion note that by definition kC is the Kan extension of kS*(C). Clearly S*(B) is a full unidense unisimple subcategory of S*(C), thus kS*(C) is trivially the Kan extension of kS*(B). It follows that kC is the Kan extension of kS*(B). By definition kB is the Kan extension of kS*(B). This implies that kC is the Kan extension of kB.

Definition 5. A functor T from C to the category of sets is called a unifunctorif the following conditions are satisfied: (a) T(X) is empty iff X is an initial object. (b) For any element p of T(X) there is a map t: P --> X in A such that T(P) has only one element and T(t)(T(P)) = p. (c) For any two non-initial maps f: P --> X and g: Q --> X in A such that T(f)(T(P)) = T(g)(T(Q)) there are non-initial maps u: O --> P and v: O --> Q such that fu = gv.

Remark 6. (a) If C is atomic then kC is a unifunctor. (b) One can show easily that a category C is atomic iff it carries a functor unifunctor T to the category of sets and T is equivalent to its unifunctor kC.

Definition 7. A functor F: D --> C between atomic categories is called uniformif kCF is equivalent to kD.

Remark 8. (a) Any unifunctor is uniform. (b) A uniform functor between atomic categories is nondegenerate. (c) A composite of uniform functors between atomic categories is uniform. (d) If F: D®C is a uniform functor then a map f in D is unipotent iff F(f) is unipotent in C.

Definition 9. An atomic category with a faithful unifunctor is called a uniconcretecategory.

In practice almost all the natural metric sites arising in geometry have atomic categories as the underlying categories and the unifunctors as the underlying set-theoretic functors for the metric topologies (but the topologies may vary). Thus in these cases the "underlying structures" are intrinsic. This is perhaps a starting point of categorical geometry. Here are some examples:

Example 9.1. Suppose C has a terminal object 1 and 1 is unidense in C. Then 1 is unisimple and C is atomic with homC(1, ~) as the unifunctor; furthermore if C is reduced then it is also uniconcrete (cf. Reduced Category) This covers many natural atomic categories, such as the left categories of sets, topological spaces, posets, coherent (i.e. spectral) spaces, Stone spaces. In fact, all of these categories are reduced, therefore uniconcrete.

Example 9.2. (a) The opposite of the category FAlg/k of finitely generated algebras over a field k is atomic. (b) The category of affine algebraic varieties over a field k is isomorphic to (FAlg/k)op, thus is atomic. (c) The category of reduced affine algebraic varieties over an arbitrary closed field is a reduced category, therefore alsouniconcrete.

Example 9.3. The category of locales is not atomic.

Example 9.4. Denote by CRing the category of commutative rings (with unit and unit preserving homomorphisms). A ring is unisimple in CRingopiff it has exactly one prime ideal (thus any field is unisimple). It is easy to see that the class of fields is unidense in CRingop. Thus CRingop is atomic (but not uniconcrete). Since the category ASch of affine schemes is equivalent to CRingop, ASch is also atomic.

Example 9.5. (a)Since ASch is a full unidense (in fact, a dense) subcategory of the category Sch of schemes, it follows from (2.c) and (9.4) that Sch is atomic. (b) A ringed space is unisimple iff its underlying space is a one point set. Since the class of such ringed spaces is unidense, the category RSpa of ringed spaces is atomic. Similarly the category LSpa of local ringed spaces is atomic.

Remark 10. In an atomic category C the unifunctor kC plays the traditional role of underlying functor:(a) A map f is unipotent iff kC(f) is surjective (by (Remark 8.d)). (b) A mono f: U -->X is normal iff kC(f): f(U) --> f(X)is an embedding (i.e. an effective mono for kC, where kCis viewed as a functor to the category of discrete spaces). For instance in an atomic analytic category (or any atomic category with a strict initial object and finite limits, see Reduced Categories) one can define the notion of a reduced objectin terms of kC: an object is reduced if any surjective map f (which means kC(f) is surjective) is epi.